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Adding a nontrivial term composed from a microstructure, we prove theexistence of a global-in-time weak solution, whose enstrophy is bounded for allthe time, to an incompressible 3D Navier-Stokes-Fourier system for arbitraryinitial data. It cannot be expected to directly derive the energy inequalityfor this new system of equations. The main idea is to employ the hydrodynamiclimit from the Boltzmann equation with periodic velocity and a speciallydesigned collision operator.

Anomalous smoothing effect on the incompressible Navier-Stokes-Fourier limit from Boltzmann with periodic velocity